Can an AI be trained to generate the outline of a story?
As far as I am aware, this has not been done yet.

I see several problems with this. A neural network is basically a classifier, which matches an input to an output. Both input and output are usually numerical values, though they could be matched to concepts or words.

To train a NN you provide an appropriately encoded input, and the corresponding output. The NN learns the associations between the two, and can then classify unseen input accordingly. This has recently been used to transform images in a particular style etc.

What would the input and output be for generating screenplays? You could use previous scripts as inputs, but what would the output be? It could be narrative 'moves' of some sort, perhaps. So you could train an NN to recognise narrative elements from screenplays.

However, you are still not creating anything, but just recognising stuff. You would need some other input. I guess you could train an NN on "The Simpsons", get a narrative structure, and then present it with an Episode of "Friends" and see what happens. It won't be a new episode of a screenplay, though.

The other way round might work: you feed it narrative moves (a kind of story skeleton), and get a script out. But it would need a lot of (human) post-editing to be at all useful.

I think an NN is the wrong tool to use here. There has been work done with generating stories and screenplays, even way back in the early days of AI. But that was all based on symbolic AI, not on the kind of ML which seems to currently be en vogue. Have a look at James Ryan's website; he has recently written an overview over historic approaches to story (and screenplay) generation.